Efficient parametric amplification in double-Λ systems without maximal two-photon coherence

We study four-wave mixing in a double-Λ system, for cw laser beams whose transverse intensity profiles (TIP’s) are initially Gaussian, by solving the Maxwell-Bloch equations numerically. For systems where coherent population trapping (CPT) is initially absent, we show that efficient frequency conversion, without focusing or ring formation, can occur even at distances which are much shorter than those required to establish CPT. We also show, for certain configurations, that blue-detuned beams become focused on propagation so that very high frequency conversion, even exceeding 100% at the center of the profile, can occur. This focusing is, however, accompanied by ring formation. We show that focusing, without ring formation, can occur for identical blue-detuned beams when the initial relative phase is π so that CPT cannot be established on propagation. The behavior of the TIP’s of the beams on propagation is explained by considering the effective linear and third-order contributions to the off-diagonal density-matrix elements.